Abstract: We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements.

The hypercube representation of a polynomial was elaborated in order to meet the requests of massively parallel computation on the "Connection Machine" at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.